A comparative simulation study on three lattice systems for the phase separation of polymer-dispersed liquid crystals

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1 PRAMANA c Indian Academy of Sciences Vol. 71, No. 3 journal of September 2008 physics pp A comparative simulation study on three lattice systems for the phase separation of polymer-dispersed liquid crystals Y J JEON 1, M JAMIL 1,2,, HYO-DONG LEE 3 and J T RHEE 2 1 Liquid Crystal Research Center, Konkuk University, Seoul , Korea 2 IAP, High Energy Physics Laboratory, Department of Physics, Konkuk University, Seoul , Korea 3 Korean Mingok Leadership Academy, Hoengseong-gun, Gangwon-do, , Korea Corresponding author mjamil@konkuk.ac.kr; jtrhee@konkuk.ac.kr MS received 15 December 2007; revised 4 April 2008; accepted 3 June 2008 Abstract. This article reports a comparative study of the phase separation process in a polymer-dispersed liquid crystal, based on a Metropolis Monte Carlo simulation study of three lattice systems. We propose a model for the different processes occurring in the formation of polymer-dispersed liquid crystals (PDLCs). The mechanism of PDLC is studied as a function of quench temperature, concentration and degree of polymerization of liquid crystals and polymers. The obtained resultant phase diagrams of the three systems are approximated and compared with the Flory Huggins theory, and show a good agreement. It has been observed in the simulation results that among all the three systems, the lattice showed the most accurate, reliable and stable results. Keywords. Monte Carlo simulation; liquid crystals; polymers; Flory Huggins theory. PACS Nos pq; Hn 1. Introduction In recent years polymer-dispersed liquid crystals (PDLCs) have received great attention because of their leading contribution to electro-optical devices manufacturing [1,2]. However, these materials still present some challenging problems related to their formation and operation. Indeed, depending on fabrication parameters, e.g. time scale, the mixtures of monomers and liquid crystals can order themselves into different microstructures either by phase separation or by domain morphology. The key product application known as polymer-dispersed liquid crystal is obtained by phase separation process, which consists of micron-sized droplets of low weight liquid crystals (LC) dispersed in a solid polymer matrix [3]. These materials can be produced by three different commonly used techniques, namely, (i) by 559

2 Y J Jeon et al polymerization-induced phase separation (PIPS) [4 6] of a polymerizing monomer mixed with a nematic liquid crystal; (ii) by thermally induced phase separation (TIPS) process [7 11] from a mixture of polymers and nematic liquid crystals; (iii) by evaporation of a common solvent dissolving the liquid crystal and polymer known as SIPS [8]. Additionally, there is another method known as reaction-induced phase separation (RIPS) [12]: phase separation performed when the polymer is diluted in an anisotropic solvent, such as liquid crystals. TIPS process is considered as a cheap, easy to handle and convenient method, while the PIPS process is useful when pre-polymer materials are miscible in low molecular weight liquid crystals. It is interesting to note that the phase separation technique used for producing PDLC films have many advantages over other methods used for manufacturing similar films [13]. This technique, applicable to a wide range of polymers, is quite useful for controlling the size and uniformity of LC droplets. Moreover, its comparatively low production cost and simplicity are additional merits. The electro-optical properties of PDLCs can be determined by the droplet size, morphology and uniformity. The phase separation of the PDLCs mainly depends on parameters like temperature, concentration and polymerization rate (in case of PIPS), cooling rate (in case of TIPS) and evaporation rate (in case of SIPS). It must be noted that the mechanism involved in these kinds of phase separation processes is due to the competition between two contributions to the available free energy. The separated state has lower entropy. The entropy is lower because the molecules of the same kind often interact between themselves more than with the molecules of other type. In the case of homogeneous state it has higher entropy. At lower temperature, energy plays a more important role than entropy, but at higher temperature the situation is the reverse. Applying these postulates, the phase transition can be predicted at some intermediate temperature, and this gives the frame of the TIPS process. In the case of PIPS, the minimum of entropy is reached when polymerizing the polymers as the bounded polymers have lower degree of freedom. The decrement of entropy is much larger in the case of homogeneous state than that of separate state. In recent years many groups have worked on such systems: similarly on different phenomenological continuous models [5,9,14 16], which were utilized in the coupling of the Landau de Gennes and Flory Huggins free energy densities with the same microscopic techniques. Some groups utilized the interface between the phases of the separated system, advances in the structures and analysed their distribution using some simple models [17]. In literature one can find simulation of some two-dimensional (2D) models [18] too. A 2D model forms a topological barrier through which other molecules cannot cross, which makes a proper simulation difficult. Therefore, we chose a 3D model for the present comparative simulation work. In this article we study the phase separation process induced by polymerization during the formation of PDLC films. Our main objective is the qualitative understanding of the effects of polymerization rate, temperature and concentration on the resulting droplet size and uniformity. To achieve this goal we perform a Metropolis Monte Carlo simulation on three different fcc lattices to simulate the polymerization-induced phase separation along with the temperature-induced phase separation in an initially uniform and randomly distributed mixture of polymer referred here as monomers and with LC molecules as well as to have a comparison 560 Pramana J. Phys., Vol. 71, No. 3, September 2008

3 Phase separation of polymer-dispersed liquid crystals of the different lattice sizes to infer something more like scaling effects. To investigate these parameters we simulated three different lattices such as (a) system, (b) system and (c) system and compared their results. 2. Simulation set-up For computing time concern, the present simulation was performed on a threedimensional model, which is discrete in space and continuous in orientation. The present simulations were done by the Metropolis Monte Carlo method. This model was first proposed by Nicholas Metropolis [18,19] and coworkers in They proposed a new sampling procedure which incorporates the temperature of the system. This is done so that the Boltzmann average property of the system can be calculated easily. Presently this modified Monte Carlo method is known as Metropolis Monte Carlo simulation method. The model we utilized to investigate the phase separation processes in a binary mixture of low-molecular weight liquid crystal molecules and polymerizing monomers is a face-centred cubic (fcc) lattice with each side occupied by either an LC molecule or a monomer. In this model, the particles were allowed to move from site to site on the fcc lattice. The fcc lattice was chosen because it is closely packed with local hexagonal structure. The simulation were carried out on (a) lattice, (b) lattice and (c) lattice. We point out that some testing calculations have been done on D lattices, which essentially gave results similar to those of the three lattices. This indicates that our system is large enough and reliable for the present studies. The number of Monte Carlo sweeps of 300,000 per molecule was chosen before reaching the thermal equilibrium from the randomly selected initial state of the system. Each trial of simulation was randomly selected among the nematogenic molecules, translation of molecules and reptation of polymer chains. The potential model used in this study can be explained in the following way: each lattice node hosts one particle: either rods (liquid crystals) or spheres (polymer), in hexagonal fcc simulation. The particles are allowed for intermolecular hard cores. Moreover, all the interactions were restricted to nearest neighbours, i.e. and U mm = b 1 ε U ll = b 2 b 3 (n j n k ) 2 ε Ul m = b 4 ε, with holes have no interactions. (1) Here n j denotes unit vectors defining particle orientations, b 1 to b 4 are positive numbers, and any of them can be set to 1, ε is a positive quantity setting energy and temperature scales in Kelvin, i.e. T = k B T/ε. The interaction between the monomer molecules is isotropic and its energy is given by ε mm = ε, for neighbour monomers, = 0 other wise. (2) Pramana J. Phys., Vol. 71, No. 3, September

4 Y J Jeon et al Another condition was set that no more than one particle is allowed to occupy the same place in the same lattice. The interaction between nematogenic molecules was similar to Lebwohl Lasher [20], i.e. ε ll = ε[b 1 + b 2 (n 1 n 2 ) 2 ]. The constants b 1, b 2, b 3 and b 4 are positive quantities which have different values for the system being investigated. By changing b 2 value one can change the temperature of nematic isotropic phase transition. The potential between the same kinds of particles is isotropic but is weaker than the mentioned two potentials, i.e. ε ml = b 4 ε. The reason is that there is a competition between energy and entropy, i.e. energy of the separate state is always lower. In order to simulate the continuous model in a better way, some lattice sites were set empty. The empty sites break the local structure and change the number of neighbour interacting particles. The number of these sites do not affect the simulation results substantially if it is properly chosen. For this reason the total number of molecules were chosen to be 64,000 in a bulk system and 63,000 in a confined system of the same size. Similarly, 27,000 was chosen as the total number of molecule in a bulk system and 26,160 in a confined system of the same size. In the case of bulk system, the total number of molecules is 125,000, and 121,110 in a confined system of the same size. The present simulation studies were made simpler by setting all the simulation parameters and results in dimensionless form. The distance between neighbour sites was measured in the length unit and the potential between adjacent monomers ε as the energy unit. The temperature unit ε/k B was inserted such that the Boltzmann constant k B = 1. Let us now proceed to understand to what extent our simulations can yield comparable results. 3. Simulation results and discussions 3.1 Validation of our model in the case of a nematogenic system As the first step of our simulation studies, the obtained results were compared with the known theoretical and simulation models. Figure 1 shows the variation of the scalar order parameter of the nematogenic system with respect to temperature, for all the three systems defined above. In such systems the interaction is similar to Lebwohl Lasher model, but it is not exactly LL itself. It is in agreement with the Landau de Gennes theory [21] and with the hexagonal Lebwohl Lasher simulation models. Figure 2 represents the dependence of the nematic correlation length ξ on temperature. This was obtained by introducing external anchoring surface having different orientational orders for all the three lattices. The correlation length is the distance from a point beyond which there is no further correlation of a physical property associated with that point. At a distance beyond this correlation length, values of a given property are considered purely random. For this figure the simulation was done on all the three confined systems. They all show the same behaviour, i.e. decrease of order parameter with increasing distance from it. Similarly, a good matching with the Landau de Gennes theory (for all the three systems) can be observed. 562 Pramana J. Phys., Vol. 71, No. 3, September 2008

5 Phase separation of polymer-dispersed liquid crystals Figure 1. A comparison of three different lattices (a) , (b) , (c) showing the nematic scalar order parameter S (i.e., the average of the second-order Legendre polynomial of the cosine angle between the molecules and director) vs. temperature. The results are fitted with the Landau de Gennes theory, which is shown as the continuous line. Figure 2. Nematic correlation length ξ vs. temperature dependence and their comparison with Landau de Gennes theory predictions (solid line) for the three syetems (a) , (b) and (c) LC polymer mixture: Influence of the degree of polymerization It is interesting to note that even if we have used different lattice systems with different ratio of nematic and monomer molecules, we have obtained similar responses. The two parameters are: the degree of polymerization and the polydispersity. The degree of polymerization is defined as the average number of monomers in a Pramana J. Phys., Vol. 71, No. 3, September

6 Y J Jeon et al Figure 3. Polydispersity coefficient U and its dependence on the average degree of polymerization N at different temperatures, for the three lattices. polymer. It is a measure of molecular weight. The polydispersity can be defined as the molecular weight non-homogeneity in a polymer system, i.e., it is the measure of molecular weight distribution throughout the body of the system. The polydispersity coefficient U is the deviation from the mean value. The effect of the degree of polymerization on the polydispersity coefficient U for different temperatures can be clearly viewed in figure 3, in which all the three systems are shown. It indicates that temperature has no effect on the distribution that was obtained as a result of polymerization procedure except for T = 1.5. Therefore, in all these systems, the number of monomer distributions with the same mean value of U are independent of temperature and degree of polymerization. To investigate the behaviour of the mixture in the bulk, the separation correlation length [22,23] ξ m and ξ lc were evaluated. According to [22], Γ(R) denotes the correlation function such that µ n = j k R n ijγ(r jk ), (3) where n denotes a non-negative integer, and one thinks of coordinates scaled by the lattice constant; j and k denote lattice sites. Then the correlation length can be defined as ξ = (µ 2 /2dµ 0 ) (4) for a d-dimensional lattice. In figure 4 the temperature-induced phase separations for the three nonpolymerized systems were observed and plotted. These systems are: (a) 8438 nematogenic and 17,720 monomer particles on lattice; (b) 21,000 nematogenic and 43,000 monomer particles on a lattice and (c) 39,060 nematogenic and 82,030 monomer particles on a lattice. In these three systems a homogeneous separated phase transition was noticed at the temperature 564 Pramana J. Phys., Vol. 71, No. 3, September 2008

7 Phase separation of polymer-dispersed liquid crystals Figure 4. Separation correlation lengths during the temperature-induced phase separation of the three systems (a) 8438 nematogenic and 17,720 monomer molecules, (b) 21,000 nematogenic and 43,000 monomer molecules, (c) 39,060 nematogenic and 82,030 monomer molecules. The phase transition between separate and homogeneous states occurs at temperature T c = T c = 0.83, with units k B = 1 and ξ mm = ±1. We point out here that the value of T c is lower than the nematic isotropic transition temperature (T NI = 1.43) observed in figures 1 and 2. This scenario indicates that at lower temperature T, the systems will be separated, while at higher temperature the systems will become homogeneous. This behaviour is due to competition between entropy and the free energy term. The sudden drop of separation correlation length indicates the transition occurrence from one state to another. Similar to figure 4, the melting of average nematic order S is observed according to the temperature T in figure 5. The average nematic order S was considered rather than the order parameter S in the nematic rich phase. The PIPS can also be characterized by other properties. In figure 6, the correlation length ξ lc of the nematic molecules during PIPS process is plotted vs. the polymerization degree N. The correlation length describes the size of the domains of single phase in the separated system. It can be observed that above the temperature T = 1.12 the three systems are in homogeneous state independently of the degree of polymerization. Conversely, below the same temperature (T s = 0.80), all the systems are in separate state. This behaviour can be well explained by the Flory Huggins theory [24,25] in its modified form. This theory is a mathematical model of the thermodynamics of polymer solutions taking into account the dissimilarity in molecular sizes, on adopting the usual expression for the entropy of mixing. Finally, the result is deduced from an equation for the Gibbs free energy change G m in the case of polymer mixed with solvent. This theory generates useful results for interpreting different experiments. Pramana J. Phys., Vol. 71, No. 3, September

8 Y J Jeon et al Figure 5. Melting of the nematic scalar order S, indicating the point where a transition to the homogeneous state occurred. Figure 6. Separation correlation length for nematic molecules ξ lc at different temperatures during the polymerization-induced phase separation of (a) , (b) , (c) lattice systems with the respective number of monomer and nematogenic molecules. A closer look at these systems show a separation process by nucleation and growth indicated by the sharp increase of the correlation length. It can be noticed that below T = 0.80 the system is always in the separated state and above this temperature (i.e., T = 1.20 and T = 1.50) always in the homogeneous one. As can be seen in figure 6, all the three systems have similar behaviour; and so here we have simulated only the system lattice. The simulated phase diagram of the system is shown in figure 7, and compared to the Flory Huggins theory [24,25]. 566 Pramana J. Phys., Vol. 71, No. 3, September 2008

9 Phase separation of polymer-dispersed liquid crystals Figure 7. Phase diagram of the system of 21,000 nematogenic and 43,000 monomer molecules in the lattice. The first curve indicates the Flory Huggins theory, while the vertical line shows the temperature where PIPS can occur. In order to perform this comparison, we made some simplification in the models in order to get the analytical results. We manually picked up the points from figure 6, where the phase transition has actually occurred, and the system could be possibly distinguished into separated and homogeneous states. By utilizing a simple model, the phase diagram of these systems was evaluated and compared with the Flory Huggins theory. The Flory parameter χ was evaluated: it has two contributions, one interactional and one conformational. χ = χ s + T χ /T. (5) The first term represents entropy and the second term can be defined as the energy contribution to χ. The small mismatch between simulation and theory could be due to the fact that the latter holds for monodisperse systems, which is different from the present system. Some interesting features with the higher concentration of nematogenic molecules resulted in the loss of its sharpness from the separated homogeneous phase transition (figures 8 and 9). The gradual loss of sharpness can be clearly seen with the decrease in temperature (in all the systems of figure 8), when the concentration of nematogenic molecules is increased. This behaviour is similar to the Flory Huggins theory which describes about a system enriched with polymer, that becomes separated in nucleation in a quick fashion. The systems with higher number of polymer molecules are considered stable with homogeneous state, and they require some condensation nuclei to become actually separated. On the other hand, the systems with low amount of polymer molecules are locally unstable and exhibit continuous change of external parameters. Similarly, figure 9 shows the gradual change in the correlation length when increasing the degree of polymerization. For the observation of PIPS, e.g., figure 9a, for lattice: (A) N lc = 4219, N m = 21,937 Pramana J. Phys., Vol. 71, No. 3, September

10 Y J Jeon et al Figure 8. Separation correlation lengths of three systems with their respective number of nematogenic and monomer molecules; showing a spinodal decomposition process indicated by a continuous change of the separation correlation length. Figure 9. The PIPS process for (a) , (b) and (c) lattices. The figures are obtained by taking different ratios of nemetogenic and monomer molecules at different temperatures where the polymerization has been performed. at T = 0.85; (B) N lc = 8438, N m = 17, 718 at T = 0.96; (C) N lc = 13,080, N m = 13,080 at T = 1.0; (D) N lc = 17,718, N m = 8438 at T = 1.10; and (E) N lc = 21,937, N m = 4219 at T = Similarly, for the other lattice sizes nematogenic and monomer molecules were taken into account. 568 Pramana J. Phys., Vol. 71, No. 3, September 2008

11 Phase separation of polymer-dispersed liquid crystals Figure 10. Phase diagram of all the three systems showing the ratios of nematic molecules (X lc ) vs. the degree of polymerization, at temperature T = 1.0. There are three phases: homogeneous, droplet and capillary states. 3.3 Simulation of the final structure Till now from the obtained results we have seen only the phase separation phenomena. In this section we investigate the final structure of all the three systems and their morphologies. We see that at very low concentration of nematogenic molecules only homogeneous structure is observed (figure 10 for all the three systems). On further increase in concentration of nematogenic molecules in all the systems, the mixture organizes itself in nematogenic droplets. At higher concentration of nematogenic molecules under certain conditions, this phase is changed into capillary structure (especially above T = 0.5). Further increase of nematogenic molecule concentration results in the formation of two structures, i.e. capillary and droplets phases. Other effects have strong impact on the structure formation, namely the internal stress in the monomers and the orientational distribution of the nematic molecules at the interface. We investigated these effects by simulating the three systems: (a) 21,931 nematogenic and 4219 monomer molecules in the lattice; (b) 52,000 nematogenic and 10,000 monomer molecules in the lattice and (c) 101,600 nematogenic and 19,531 monomer molecules in the lattice (figure 11). All the three systems are first polymerized and later the temperature of the systems is gradually modified. At low temperature all the systems keep orientational order and the polymer phase shows fibre-type of the structure. Then at the critical temperature (T = 1.3), the fibre structure is transformed into the polymer droplet structure. However, if the temperature is still increased (above T = 1.7), the systems are transformed into the homogeneous state. Pramana J. Phys., Vol. 71, No. 3, September

12 Y J Jeon et al Figure 11. Correlation between the fibre-to-droplet and droplet-to-homogeneous state transitions. Here the temperatures between the two transitions coincide. Actually in the thermally-induced phase separation (TIPS), this behaviour occurs when the polymer binder has a melting temperature below its composition temperature. In this PDLC separation method, a homogeneous mixture of liquid crystal and a melted polymer is formed. As it has been seen till now that all the three systems behave similarly, for simplicity we focus on the lattice system having 43,000 monomer and 21,000 nematogenic molecules. A closer look into the system shows that at the beginning (temperature T = 0.7) the system is in homogeneous state. As we increase the temperature (T = 0.8) the system begins to segregate and the droplets of liquid crystals and polymers can be seen. On further increasing the temperature of the system (T = 0.83), the system is fully separated and the droplet sizes of liquid crystal and polymer grow up in size. Further increase of temperature does not contribute in TIPS formation and the system under study returns back to homogeneous state. The corresponding pictures can be seen in figure Conclusions In order to study the phase separation processes of nematogenic and polymer molecules in mixtures, we have performed Monte Carlo simulations on three different system lattices. We have particularly examined the phenomena of phase separation as a function of temperature, polymerization and concentration of nematogenic and polymer molecules. On simulating these three systems, first all the systems were generated by an appropriate number of nematogenic and monomer molecules without temperature, thus the systems were non-thermalized systems. Secondly they were thermalized to some specific temperatures. In the third step these lattices were set for polymerization in the respective systems. In some cases the systems were polymerized two or three times. Finally, the physical quantities like order parameter, the nematic correlation lengths etc., were calculated. From the simulation 570 Pramana J. Phys., Vol. 71, No. 3, September 2008

13 Phase separation of polymer-dispersed liquid crystals Figure 12. A schematic diagram of the lattice system showing the TIPS separation process. (a) The system is in homogeneous state at T = 0.7, (b) the separation of nematogenic and monomers has started at T = 0.8 and the droplets started growing, (c) the system is completely separated and transition has occurred, where the yellow green particles are monomers and the blue droplets indicate the nematogenic molecules, (d) on further increasing temperature the system does not show transition any longer and the system returns back to homogeneous state. results and the morphology of the configuration of the systems, we observed that the present results are in good agreement with the Flory Huggins theory. We found the second system (i.e lattice), as the most reliable, compact and stable among all the three systems. We have also investigated the temperature-induced phase separation process in such systems. It was observed that in the mixture of polymer and LC molecules, at sufficiently high temperature, from the homogeneous state droplet of LCs and polymers are formed. The droplets then become larger in size. Finally a stage comes where the system turns back into homogeneous state. The authors do bear in mind that the sizes of the systems and their dimensionalities also play a vital role in this kind of simulation work as different sizes and dimensionality might give different behaviour and results. Acknowledgements We are grateful to Matej Bazec, University of Ljubljana, Jadranska Ljubljana, Slovenia, Bob Bellini, National Research Council Institute for Microelectronics and Microsystems (CNR-IMM), Rome, Italy and Prof. Silvano Romano from Pramana J. Phys., Vol. 71, No. 3, September

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